Theorem bj-eldiag 38065

Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 38059. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-eldiag	⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))

Proof of Theorem bj-eldiag

Step	Hyp	Ref	Expression
1		bj-diagval2 38064	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
2	1	eleq2d 2847	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ 𝐵 ∈ ( I ∩ (𝐴 × 𝐴))))
3		elin 3915	. . 3 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)))
4		ancom 466	. . 3 ⊢ ((𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ))
5		bj-elid4 38057	. . . 4 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 ∈ I ↔ (1st ‘𝐵) = (2nd ‘𝐵)))
6	5	pm5.32i 585	. . 3 ⊢ ((𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))
7	3, 4, 6	3bitri 300	. 2 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))
8	2, 7	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   I cid 5545   × cxp 5649  ‘cfv 6531  1^st c1st 7988  2^nd c2nd 7989  Idcdiag2 38061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6487  df-fun 6533  df-fv 6539  df-1st 7990  df-2nd 7991  df-bj-diag 38062

This theorem is used by: (None)